📐 geometry

1. **State the problem:** We are given two sides AB and BC of a regular polygon with $n$ sides, and two angles at vertex B: $148^\circ$ and $50^\circ$. We need to find the number o

1. **State the problem:** We have a triangle with angles labeled $y$, $x$, $50$, $70$, and $z$. We are asked to find the value of $x$. 2. **Recall the triangle angle sum rule:** Th

1. **State the problem:** We have a triangle with angles labeled 12°, 50°, and 70°, and we need to find the values of $x$, $y$, and $z$ which presumably correspond to these angles

1. **State the problem:** We have a trapezoid with height $h = 11$ cm, area $A = 550$ cm², and one base $b_1 = 10$ cm. We need to find the length of the other base $b_2$. 2. **Form

1. The problem asks to complete the table by measuring the side lengths of triangles △ABC and △DEF, then calculate the ratios $\frac{AB}{DE}$, $\frac{BC}{EF}$, and $\frac{AC}{DF}$.

1. **Stating the problem:** We are asked to find the height of a tree using indirect measurement. A 4-meter flagpole casts a 5-meter shadow, and a nearby tree casts a 20-meter shad

1. **Problem 4:** Given angles around a circle: 137°, $x^\circ$, 223°, and 111.5°. We need to find $x$. 2. The sum of angles around a point on a circle is 360°.

1. **State the problem:** A carpenter claims that in a 30°-60°-90° triangle with hypotenuse 12 m, the other two sides must be 6 m and 10.39 m. 2. **Recall the properties of a 30°-6

1. **State the problem:** We are given two triangles, \(\triangle ABC\) and \(\triangle DEF\), with \(\angle A \cong \angle D\). The sides including these angles are \(AB = 6\) cm,

1. The problem asks us to verify the Pythagorean Theorem for a right triangle with legs 6 cm and 8 cm. 2. The Pythagorean Theorem states that in a right triangle, the square of the

1. **Problem statement:** Find the size of angle $XYZ$ in a right triangle with right angle at $X$, side $XZ = 13$ cm, and side $XY = 4$ cm. 2. **Identify the sides relative to ang

1. **Problem statement:** Find the size of angle $XZY$ in a right triangle $XYZ$ where $\angle X = 90^\circ$, side $XY = 6$ cm, and side $ZY = 15$ cm. 2. **Identify the sides relat

1. **State the problem:** We are given a quadrilateral PQRS inscribed in a circle with the following conditions: - $m\angle R = m\angle S$

1. **Problem Statement:** Quadrilateral PQRS is inscribed in circle A. Given that $m\angle P = 92^\circ$ and $m\angle Q = 92^\circ$, determine which statement about angles $R$ and

1. **State the problem:** We have a circle with center $O$ and points $A, B, C, D$ on the circumference.

1. **State the problem:** Determine if the hexagon JNLKOM constructed by Veronica is a regular hexagon inscribed in circle C. 2. **Recall definitions and properties:**

1. The problem asks us to evaluate the reasoning of two students proving the Pythagorean Theorem for a right triangle with legs 5 cm and 12 cm. 2. The Pythagorean Theorem states th

1. **State the problem:** We have a flagpole casting a 6 m shadow and a 2 m tall post casting a 1.5 m shadow at the same time. We want to find the height $h$ of the flagpole using

1. **Stating the problem:** We have a rectangle cut along a dotted line forming two parts, P and Q. We want to find how the perimeter of Q relates to the perimeter of P. 2. **Under

1. **State the problem:** We are given a solid cylinder with radius $r = 8$ cm and volume $V = 3892$ cm³. We need to find the height $h$ of the cylinder, correct to one decimal pla

1. **Stating the problem:** Identify which pairs of angles among \(\angle 5 \text{ and } \angle 4\), \(\angle 6 \text{ and } \angle 5\), \(\angle 3 \text{ and } \angle 5\), and \(\